
(REF ) is included to facilitate comparison with Theorem 2.10 of [1]}. In the regime where \(\beta _n \rightarrow \beta > 0\)  and \(\theta _i > 1\)  is constant, the term multiplying \(|\langle \tilde{v}_i, v_i \rangle |^2\)  converges to a positive limit, and \(|\langle \tilde{v}_i, v_i \rangle |^2\)  converges to the limit's inverse. Here, since
\( \phantom{}\Big |\theta _i^2 | \langle \tilde{u}_i, u_i \rangle |^2 \beta _n \Big ( \bar{s}_n(\bar{\lambda }_i) + \bar{\lambda }_i \frac{d}{dz} \bar{s}_n(\bar{\lambda }_i) \Big ) \Big | \lesssim \sqrt{\beta _n} \,, \) 
